1.

If `f(x)=[{:(mx+1,if x le (pi)/(2)),(sinx+n,ifxgt(pi)/(2)):}` is continuous at `x = (pi)/(2)`, thenA. `m = 1, n = 0`B. `m = (n pi)/(2)+1`C. `n = (mpi)/(2)`D. `m = n = (pi)/(2)`

Answer» Correct Answer - C
We have, `f(x)=[{:(mx+1,if x le (pi)/(2)),(sinx+n,ifxgt(pi)/(2)):}` is continuous at `x = (pi)/(2)`,
`:. LHL = underset(xrarr(pi^(-))/(2))(lim)(mx+1)=underset(hrarr0)(lim)[m(pi/2-h)+1]=(mpi)/(2)+1`
and `RHL = underset(xrarr(pi^(+))/(2))(lim)(sinx+n)=underset(hrarr0)(lim)[sin(pi/2+h)+n]`
`= underset(hrarr0)(lim)cosh+n=1+n`
`:. LHL = RHL` , [to be continuous at `x = (pi)/(2)`]
`rArr m.(pi)/(2) + 1 = n + 1`
`:. n = m.(pi)/(2)`


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